The dynamics of the gravitational two-body problem in the post-Newtonian approximation at quadratic order in the Newton's constant.

The two body problem in general relativity is not exactly solvable. A viable strategy for weakly gravitating systems is to perform a (so-called post-Newtonian) perturbative expansion in terms of the small parameter G (m₁+m₂) / r. Nevertheless, going up in the perturbative series becomes encreasingly difficult, and the energy of the two-body system is presently known only at the third post-Newtonian order.

In this paper we compute for the first time a definite class of terms of the fouth post-Newtonian Lagrangian, that is all the terms containing two or less powers of Newton's contant G.This result has been obtained by means of a Mathematica code based on the effective field theory methods for the gravitationally bound two-body system.